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Abstract

Long-path pulse-to-pulse interferometers of two-color frequency combs are developed using fundamental and second harmonics of a mode-locked fiber laser. Interferometric phase difference between two-color frequency combs was precisely measured by stabilizing the fundamental fringe phase by controlling the repetition frequency of the comb, and a stability of 10−10 for 1000 s was achieved in the measurement of an optical path length difference between two wavelengths. In long-term measurements performed for 10 h, results of phase variation of interferometric measurements were highly consistent with the fluctuations in the calculated difference of refractive indices of air at two wavelengths with an accuracy of 10−10. The difference between the measured optical distances corresponding to two wavelengths and the optical distance corresponding to the fundamental wavelength were used in the two-color method; high-accuracy self-correction of the fluctuation of refractive index of air was performed with an uncertainty of 5 × 10−8 for 10-h measurements when the maximum refractive index change was on the order of 10−6.

Figures (7)

Fig. 1 Experimental set up for the pulse-to-pulse interferometer using two-color optical combs. The output of the fiber comb is separated into three branches; one of them is used for stabilizing the comb (not shown) and other two are used for interferometric experiments for fundamental (ω) and SHG (2ω). EDFA: Er-doped fiber amplifier; FC: fiber collimator; H: half-wave plate; L: lens; C: SHG crystal; DM: dichroic mirror; BS: beam splitter of 10 mm cube; PD1,2: photodetectors. Path difference between the probe and reference is 5.55 m, which corresponds to the repetition rate of 54.0 MHz. In the interferometer, the pulses were highly chirped by propagation in fibers and a PPLN crystal. The pulse durations estimated by the coherence lengths were about 1 ps and 3 ps for fundamental and SHG, respectively.

Fig. 3 Environmental variation of interference fringe signals of two-color combs ((a) fundamental, (b) SHG) and (c) their difference. In (a) and (b), the origin of each vertical axis is one of the zero phases. Residual sinusoidal variation in the plot of (c) is due to the imperfect temperature control of the SHG crystal oven.

Fig. 4 Interferometric measurements of two-color frequency combs. The fundamental interference fringe signal (a, b) is stabilized by feedback to frep. Variations of the fundamental and SHG optical path lengths are shown ((a, b) and (c, d), respectively). The origin of vertical axis is zero phase of the fundamental fringe. Moving average of 25 s was applied in the cases of (b) and (d).

Fig. 5 Long-term variation in the difference between the refractive indices of air at two color wavelengths n2-n1 (a) obtained by the measured phase difference of the two-color interferometer, (b) calculated using environmental parameters, and (c) their difference. The vertical positions of the plots were shifted for clarity as follows. In (a) and (b), the vertical origins of the plots was chosen so that the averages of all the data in two plots were the same. In (c), the vertical origin was shifted so that the average of data was 2x10−9.

Fig. 6 Long-term variation in the refractive index of air (a) obtained by the measured frep, (b) calculated using environmental parameters, and (c) their difference. The origin of each plot was arbitrary chosen for clarity in the similar way to Fig. 5. In (c), the vertical origin is shifted so that the average is 0.2 × 10−6.

Fig. 7 Self-correction of air-refractive index by two-color method. The variations in the relative distances to the target distance D, which are obtained from the corresponding refractive indices, are plotted; (a) one-color distance D1 obtained by measured frep (Fig. 6(a)), (b) difference between optical distances corresponding to two colors, D2 - D1, obtained from the direct measurements of phase difference of the two-color interferometers (Fig. 5(a)), and (c) corrected geometrical distance D obtained using Eq. (3). The origin of each plot was arbitrary chosen for clarity in the similar way to Fig. 5. The vertical origin of the plot (c) was shifted so that the average of data is 0.2 × 10−6.